Chaotic diffusion caused by close encounters with several massive asteroids: II. the regions of (10) Hygiea, (2) Pallas, and (31) Euphrosyne

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DOI:10.1051/0004-6361/201220448

c

ESO 2013

Astrophysics

&

Chaotic diffusion caused by close encounters with several

massive asteroids

II. The regions of (10) Hygiea, (2) Pallas, and (31) Euphrosyne

⋆

V. Carruba

1

, M. Huaman

1

, R. C. Domingos

2

, and F. Roig

3

1 _{UNESP, Univ. Estadual Paulista, Grupo de dinâmica Orbital e Planetologia, Guaratinguetá, SP, 12516-410, Brazil}

e-mail:vcarruba@feg.unesp.br

2 _{INPE, Instituto Nacional de Pesquisas Espaciais, São José dos Campos, SP, 12227-010, Brazil}

3 _{ON, Observatório Nacional, Rio de Janeiro, RJ, 20921-400, Brazil}

Received 26 September 2012/Accepted 12 December 2012

ABSTRACT

Context.Close encounters with (1) Ceres and (4) Vesta, the two most massive bodies in the main belt, are known to be a mechanism of dynamical mobility able to significantly alter proper elements of minor bodies, and they are the main source of dynamical mobility for

medium-sized and large asteroids (D>20 km, approximately). Recently, it has been shown that drift rates caused by close encounters

with massive asteroids may change significantly on timescales of 30 Myr when diﬀerent models (i.e., diﬀerent numbers of massive

asteroids) are considered.

Aims.So far, not much attention has been given to the case of diﬀusion caused by the other most massive bodies in the main belt: (2) Pallas, (10) Hygiea, and (31) Euphrosyne, the third, fourth, and one of the most massive highly inclined asteroids in the main belt, respectively. Since (2) Pallas is a highly inclined object, relative velocities at encounter with other asteroids tend to be high and

changes in proper elements are therefore relatively small. It was thus believed that the scattering eﬀect caused by highly inclined

objects in general should be small. Can diﬀusion by close encounters with these asteroids be a significant mechanism of long-term

dynamical mobility?

Methods.By performing simulations with symplectic integrators, we studied the problem of scattering caused by close encounters with (2) Pallas, (10) Hygiea, and (31) Euphrosyne when only the massive asteroids (and the eight planets) are considered, and the other massive main belt asteroids and non-gravitational forces are also accounted for.

Results.By finding relatively small values of drift rates for (2) Pallas, we confirm that orbital scattering by this highly inclined object

is indeed a minor eﬀect. Unexpectedly, however, we obtained values of drift rates for changes in proper semi-major axis acaused

by (10) Hygiea and (31) Euphrosyne larger than what was previously found for scattering by (4) Vesta. These high rates may have repercussions on the orbital evolution and age estimate of their respective families.

Key words.minor planets, asteroids: general – minor planets, asteroids: individual: (10) Hygiea – celestial mechanics –

minor planets, asteroids: individual: (31) Euphrosyne – minor planets, asteroids: individual: (2) Pallas

1. Introduction

Orbital mobility caused by close encounters with massive aster-oids has been studied in the past and could be a viable mech-anism to produce the current orbital location of some of the V-type asteroids presently outside the Vesta family. It is well known, however, that the proper frequencies of precession of pericenterg and longitude of the node s of terrestrial planets change when one or more of the other planets is not considered in the integration scheme. For instance, theg4 ands4

frequen-cies of Mars are diﬀerent when the full solar system is consid-ered or when only Mars and the Jovian planets are accounted for. Carruba et al. (2012) showed that (4) Vesta proper frequencies are dependent on the number of other massive asteroids included in the integration scheme and that, as a result, the whole statistics of encounters with (4) Vesta is also aﬀected.

Delisle & Laskar (2012) showed that dynamical mobility caused by close encounters with massive asteroids is dominated

⋆ _{Appendix A is available in electronic form at}

http://www.aanda.org

by encounters with (4) Vesta and (1) Ceres, with the other mas-sive asteroid playing a lesser role. Previous studies on this sub-ject (Carruba et al.2003,2007) also focused on the dynamical mobility caused by the two largest bodies. Can the third, fourth, or other most massive bodies have also played a role in the dy-namical mobility of minor bodies, at least in their orbital proxim-ities? And if that is possible, how much is the drift rate in semi-major axis aﬀected by the choice of the model used to simulate the long-term eﬀect of close encounters with massive asteroids, as observed in the region of (4) Vesta (Carruba et al.2012)?

In this work we will focus on the dynamical mobility caused by close encounters with the third ((2) Pallas) and the fourth ((10) Hygiea) asteroid in the main belt, with attention on the scattering eﬀect on members of their respective fami-lies. Since one of the goals of this investigation is to study the long-term eﬀect of close encounters with highly inclined mas-sive asteroids, we also considered the case of (31) Euphrosyne. Previous studies (Baer et al.2011) have indicated that this body was the fifth-largest in the main belt. New estimates (Carry 2012) have considerably reduced its mass, but it remains one of

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the most massive bodies that are part of a significantly large as-teroid family (Machuca & Carruba2011). Since the population of large asteroids is somewhat limited in the orbital proximity of (2) Pallas, we studied the long-term orbital diﬀusion caused by close encounters with this body also to better assess how rel-evant this eﬀect is for highly inclined asteroids. In particular, if we consider the frequency distribution function of relative veloc-ities at close encounters for highly inclined asteroids, does there exist a significant statistical tail for which relative velocities are small enough that large changes in proper semi-major axisacan be observed?

This work is structured as follows: in Sect. 2 we study the orbital neighborhood of (2) Pallas, (10) Hygiea, and (31) Euphrosyne by obtaining dynamical maps. Section 3 is dedicated to estimating the changes in the proper frequencies of these massive asteroids observed when different integration schemes are considered. In Sect. 4 we obtain the statistics of changes in proper semi-major axis caused by close encounters only when different models of the main belt are considered. In Sect. 5 we discuss the statistical completeness of the frequency distribution functions that we obtained in Sect. 4. In Sect. 6 we also include in our models non-gravitational forces such as the Yarkovsky and YORP effects. Finally, in Sect. 7 we present our conclusions.

2. Dynamical maps

We start our analysis by studying the orbital regions where the massive asteroids of interest are currently located in the main belt. With a mass of 8.63 _× 1019 _{kg, (10) Hygiea is the fourth}

most massive body in the main belt. According to Milani & Kneževi´c (1994), (10) Hygiea and its family reside in a fairly stable area of the outer main belt, between the 9J:-4A and the 2J:-1A mean-motion resonances in semi-major axis. The most notable secular resonance in the area is thez1=g−g6+s−s6,

that, computed for the values of propera,e, andiof (10) Hygiea, crosses a region just above the typical inclination values of Hygiea family members.

To confirm this scenario, we obtained a dynamical map of synthetic proper elements, computed with the approach of Kneževi´c & Milani (2003), Carruba (2010), for the region of (10) Hygiea. We took 6464 particles between 2.950 and 3.265 AU in semi-major axis, 0◦ _{and 10}◦ _{and in}_i_{, with a} step-size of 0.005 AU inaand 0.1◦_in_i_{. All the other elements of the} test particles were those of (10) Hygiea at the osculating date of J2000. Our results are shown in Fig.1, where the test particles appear as black dots. Mean-motion resonances will appear in this map as vertical strips with lower densities of test particles, while secular resonances appear as elongated diagonal alignments of bodies (the names of secular resonances in the region are shown in red). Regular regions show a uniform population of test par-ticles. The position of (10) Hygiea itself is shown as a magenta full dot. We also show in the figure the orbital position of all real asteroids with absolute magnitudesH <13.5 in the samea₋i

range of our test particles as blue asterisks1_{. As in Carruba et al.} (2012), we choose only the largest bodies because asteroids with diameters larger than≃20 km are the ones for which close en-counters are the dominant mechanism of dynamical mobility on

1 _{The dynamical map was computed using the orbital elements of}

(10) Hygiea at J2000. The location of secular resonances may vary when computed for the orbits of other asteroids. See also Carruba & Morbidelli (2011).

2.95 3 3.05 3.1 3.15 3.2 3.25 3.3

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22

Hygiea

7J:−3A 9J:−4A

1J:1S:3A 2J:5S:−2A

2J:−1A

z

1

z

1

Proper semi−major axis [AU]

Sine of proper inclination

Fig. 1.Dynamical map of 6464 test particles in the region of the Hygiea

family (black dots). We note the presence of thez1secular resonance,

which appears as an elongated diagonal strip of aligned test particles,

near the Koronis family. Real asteroids in the region withH < 13.5

are shown as blue asterisks, the orbital position of (10) Hygiea itself appears as a magenta full dot.

timescales of 250 Myr, according to Delisle & Laskar (2011; see also a discussion in Carruba et al.2003). Using the relationship

D(km)=132910

(₋H/5)

√_p V

, (1)

wherepV is the geometric albedo, equal approximately to 0.05 for typical C-type objects and 0.1 for typical S-type objects (Bowell et al. 1989), we find that H = 13.5 corresponds to diameters of approximately 11.86 km for C-type objects and 8.39 km for S-type objects, respectively. These diameters are still in the regime where orbital mobility caused by close encounters is not negligible when compared with the mobility caused by the Yarkovsky eﬀect.

To avoid including Koronis family members we only se-lected sin (i)<0.14 asteroids for the Hygiea region. The Hygiea family lies in a fairly regular region, at about 0.1 in sin (i). We then considered the region of (2) Pallas. We took 5000 particles between 2.7 and 2.85 AU in semi-major axis, 25◦_{and 48}◦ _in_i_, with a step-size of 0.0025 AU inaand 0.2◦ini. All the other elements of the test particles were those of (2) Pallas at J2000. Our results are shown in Fig.2, where the test particles appear as black dots. The position of (2) Pallas itself is shown as a ma-genta full dot. We also show in the figure the orbital position of all real asteroids with absolute magnitudesH<13.5 in the same

a₋irange of our test particles.

Finally, we investigate the case of (31) Euphrosyne. We took 4500 particles between 3.02 and 3.32 AU in semi-major axis, 20◦ and 35◦ in i, with a step-size of 0.005 AU ina and 0.2◦ in i. All the other elements of the test particles were those of (31) Euphrosyne at J2000. Our results are shown in Fig.3, where the test particles appear as black dots. The position of (31) Euphrosyne itself is shown as a magenta full dot. All real asteroids with absolute magnitudesH <13.5 in the samea₋i

range of our test particles are also shown in the figure.

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2.7 2.75 2.8 2.85 0.45

0.5 0.55 0.6 0.65 0.7 0.75 0.8

Pallas

8J:−3A 5J:−2A

1J:−3S:1A

18J:−7A ν

5

ν 16

ν 5−ν16

Pallas

Fig. 2.Dynamical map of 5000 test particles in the region of the Pallas family (black dots). The orbital location of (2) Pallas is shown as a

magenta dot, the other symbols are the same as in Fig.1.

3.05 3.1 3.15 3.2 3.25

0.35 0.4 0.45 0.5 0.55

Euphrosyne

9J:−4A

1J:1S:3A 2J:5S:−2A

2J:−1A

ν

6

ν

16

ν

5

z₂

Fig. 3. Dynamical map of 4500 test particles in the region of the Euphrosyne family (black dots).

rates caused by close encounters with (2) Pallas, (10) Hygiea, and (31) Euphrosyne. This is the subject of the next section.

3. The indirect effect of other massive asteroids: choosing the right model

To investigate for possible indirect eﬀects caused by the pres-ence of other massive asteroids in the model, we obtained the values of the proper frequencies of pericenter and longitude of the node precession g and s for (2) Pallas, (10) Hygiea, and (31) Euphrosyne for simulations involving progressively all 39 main belt asteroids more massive than 8.5_× 1018 _kg

in the Carry (2012) paper2. The proper frequencies were com-puted with the frequency modified Fourier transform method of Laskar (1988,1990,1993), using an algorithm also described in Šidlichovský & Nesvorný (1997). A list of the first 39 most massive asteroids is given in TableA.1. For the mass of (4) Vesta,

2 _{The first simulation for obtaining Pallas frequencies had the eight}

planets plus (2) Pallas (S1 model), the second also included (1) Ceres (S2 model), the third included all of the above plus (4) Vesta (S3 model), and so on.

we used the value obtained from the DAWN mission (Russel et al.2012). Following the approach of Carruba et al. (2012), we integrated our asteroids as massive bodies over 30 Myr with SWIFT-SKEEL, the symplectic integrator using the Wisdom and Holman mapping (Wisdom & Holman 1991; Levison & Duncan 2000). Results for the g frequency (for simplicity we do not show thesdata) are shown in Fig. 4, panels A, C, and E. For each of the 39 simulations with massive asteroids, we computed the values of thegandsfrequencies over four time intervals of 8.192 Myr using the approach of Kneževi´c & Milani (2003). We took the mean of the four values as an estimate of the proper frequencies and the standard deviation as an estimate of the er-ror. Vertical black lines display the error associated with each frequency value, the blue horizontal line shows the mean value ofgfor all the 39 simulations, and the horizontal red lines dis-play the values of frequencies between the mean value minus and plus the standard deviation of the data (we refer to this in-terval as the “confidence level”). Panels B, D, and F of Fig.4 display the number of close encounters among massive aster-oids as a function of the number of massive asteraster-oids included in the model.

For the case of (10) Hygiea, the number of close encounters experienced by this asteroid during the simulation was some-what limited when compared with results observed for (4) Vesta (Carruba et al.2012). The models for which this number was the highest correspond to values ofg frequencies beyond a 3σ

level, but not to the highest (or lowest) peaks (models with 12 and 14 asteroids, for instance). Mechanisms other than close en-counters with massive asteroids, possibly such as secular pertur-bation of the precession frequency of (10) Hygiea by other mas-sive asteroids, should be at play to explain the diﬀerent values of secular frequencies in these cases. The first model for which the frequency was close to the mean value occurred for six asteroids, while the first peak in thegfrequency for models with more than four asteroids was observed for nine asteroids.

In the (2) Pallas case, while the number of close encoun-ters was higher than in the (10) Hygiea one, we still do not see a clear correspondence between the models with the largest number of close encounters and the peaks in thegdistribution. The largest peaks (models S16 and S28) are not associated with close-encounter events. The first model with more than four as-teroids for which the frequency was close to the mean value was the S6, while the first peak in thegfrequency was observed in the S9.

Finally, (31) Euphrosyne, more than (10) Hygiea and (2) Pallas, experienced only a very limited number of close en-counters with massive asteroids. The first model for which the frequency was close to the mean value occurred for seven aster-oids, while the first peak in thegfrequency for models with more than five asteroids (the four most massive plus (31) Euphrosyne) were observed for six asteroids.

Table1summarizes our results for the three massive aster-oids that we investigated. The second column shows the model for which the asteroidgfrequency was closer to the mean value, and the third column the model for which the first peak beyond a 3σlevel was observed. We will use these models in Sect.4to investigate the long-term behavior of chaotic diﬀusion caused by close encounters with these massive asteroids.

4. Chaotic diffusion caused by close encounters

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0 5 10 15 20 25 30 35 40 127.5

128 128.5 129 129.5

(10) Hygiea

Asteroid Number

g frequency ["/yr]

A

0 5 10 15 20 25 30 35 40

0 0.2 0.4 0.6 0.8 1

Asteroid Number

Number of encounters with (10) Hygiea

B

0 5 10 15 20 25 30 35 40

−1.55 −1.5 −1.45 −1.4 −1.35 −1.3 −1.25 −1.2 −1.15

(2) Pallas

Asteroid Number

g frequency ["/yr]

C

0 5 10 15 20 25 30 35 40

0 0.5 1 1.5 2

Asteroid Number

Number of encounters with (2) Pallas

D

0 5 10 15 20 25 30 35 40

31 31.2 31.4 31.6 31.8 32 32.2 32.4

(31) Euphrosyne

Asteroid Number

g frequency ["/yr]

E

0 5 10 15 20 25 30 35 40

0 0.2 0.4 0.6 0.8 1

Asteroid Number

Number of encounters with (31) Euphrosyne

F

Fig. 4.PanelsA),C),E)dependence of the precession frequency of the Hygiea, Pallas, and Euphrosyne pericentergon the number of other

massive asteroids included in the simulation. Asteroids are given in the order of TableA.1. PanelsB),D),F)number of close encounters among

massive asteroids as a function of the number of asteroids in the model.

the SWIFT-SKEEL code of the SyMBA package of Levison & Duncan (2000). We integrated our asteroids over 30 Myr and used the S0 simulation to estimate the noise in the change in propera caused by secular eﬀects with the method described

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Table 1.Integration schemes for which the asteroidgfrequency was closer to the mean value (second column) and for which the first peak

beyond a 3σlevel was observed.

Asteroid First mean value First peak ing

(10) Hygiea S6 S9

(2) Pallas S6 S9

(31) Euphrosyne S7 S6

asteroids will be the sum of the variance of the real diﬀusion and the variance of the noise. Therefore, the real diﬀusion resulting from close encounters will be given by

σ2a[1]=σ

2

a[1]−σ

2

noise[1]. (2)

We used the results of the S0 simulation to estimate the value of the variance caused by the noise. As in Carruba et al. (2012), we found a value of the standard deviation, the square root of the variance, equal toσnoise ≃2×10−4AU. Since the stronger

changes in propera(identified as∆ahereafter) are those that i) are more significant for the dynamical mobility caused by close encounters with massive asteroids (Carruba et al. 2003); and ii) are those that are less likely to be caused by secular eﬀects, we concentrated our attention on the values of∆a > 3σnoise =

6 × 10−4_AU.

We also computed the skewness, kurtosis higher moments of the distributions of changes in propera, and the The Hurst expo-nentT of the changes inσa[1]. The skewness gives a measure of the asymmetry of the probability distribution of a real-value ran-dom variable, while the kurtosis is any measure of the “peaked-ness” of the probability distribution of a real-value random vari-able (see Carruba et al.2012, for a more in-depth description of these parameters). We assumed that the Hurst exponent could be fitted by a power law of the time of the form

σa[1](t)=CtT, (3)

whereCis a constant. The Hurst exponent gives a measure of how much the diffusion process differs from a random walk. It is equal to 0.5 for random walk-like processes, and is between 0.5 and 1.0 for correlated and persistent processes. Previous values ofT for diffusion caused by close encounters with (1) Ceres for members of the Adeona and Gefion families were of the order of 0.58–0.72 (Carruba et al.2003). Here we compute this ex-ponent using Eq. (3) to best fitT using three values ofσa[1](t) computed at each 10 Myr. For each of our simulations, we there-fore computed the following parameters: the total number of en-counters, the number of encounters for which∆awas larger than 3σnoise, the reduced standard deviationσa[1] without the “noise”

of the simulation, obtained from Eq. (2), the skewness, kurtosis, and Hurst exponent. We start our analysis by investigating the Hygiea region.

4.1. (10) Hygiea

For the region of (10) Hygiea, we simulated the 2914 realH <

13.5 asteroids under the influence of all planets plus (10) Hygiea with a time-step of two days (see Carruba et al.2012, for a dis-cussion on the choice of this parameter), and we monitored dur-ing the simulation if the distance from this asteroid was less than 0.001 AU (see discussion in Carruba et al.2012, for the choice of this parameter). We used five integration schemes: S0, S1, S4, and the ones whose proper frequenciesg and sdiﬀered most

from the mean value (S6 and S9, respectively). As in Carruba et al. (2012), we found that the same asteroids did not experience close encounters in diﬀerent integration schemes (for instance, asteroids that experienced large changes inain one integration scheme may not have experienced close encounters at all in an-other) and that there were fluctuations in the statistics of∆aas well.

Table2summarizes our results. Histograms of frequencies of changes of proper a (Ni/Ntot, where Ni is the number of encounters in the ith bin and Ntot is the total number of

en-counters) for the four integration schemes used are also shown in Fig. 5, panel A. Unexpectedly, we found large values of

σa[1] caused by encounters with (10) Hygiea: the mean value of 90.8×10−5AU is even larger than the corresponding mean value ofσa[1] = (78.6±15.6)×10−5 _{AU caused by}

encoun-ters with (1) Ceres, the main perturber in the main belt, in the (4) Vesta region as found in Carruba et al. (2012). We also found that 12.4% of the encounters satisfied our criterion for relevance of causing a change in∆alarger than 3σ =6×10−4_{AU. This}

is by far the largest percent fraction found in our simulations so far. The unexpected high rate of drift in properacaused by en-counters with (10) Hygiea, unknown in previous works in the literature, may have played a yet to be investigated significant role in the evolution of the Hygiea family.

4.2. (2) Pallas

For the region of (2) Pallas, we simulated the 300 real H <

13.5 asteroids under the influence of all planets plus (2) Pallas, using the same integration schemes as discussed in Sect.4.1. The limited number of known objects in the Pallas region is caused by the fact that this is a highly inclined region well above theν6

andν5 secular resonances (see Carruba & Machuca2011, for a

more in-depth discussion on the highly inclined asteroid popula-tion in stable and unstable areas). We used the standard S1 and S4 integration schemes, plus the S6 and S9 schemes associated with the first mean value and the first peak in thegfrequency of (2) Pallas discussed in Sect.3. Results for the∆astatistics caused by close encounters with (2) Pallas are summarized in Table 2. A histogram of frequency of ∆a values is shown in Fig.5, panel B.

Contrary to what was found for (10) Hygiea, we found the rather low value of meanσa[1] (20.8×10−5_{AU), just a factor two}

higher than what was found for encounters with (2) Pallas in the Vesta region in Carruba et al. (2012). Only a very limited num-ber of close encounters satisfied our criterion for relevance of causing a change in∆alarger than 3σ(at most, just three for the S9 simulation). The very low value of drift in properacaused by encounters with (2) Pallas seems to confirm the hypothe-sis of Bottke et al. (1994) that encounters with this highly in-clined asteroid occur mostly at high relative speed and distances and therefore cause limited changes in proper elements. To fur-ther investigate this hypothesis, we consider the case of (31) Euphrosyne in the next sub-section.

4.3. (31) Euphrosyne

For the region of (31) Euphrosyne, we simulated the 2138

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Table 2.Numbers of encounters, numbers of encounters with|da| >3σnoise, and moments (in AU) of the distribution of changes in proper a

caused by (10) Hygiea, (2) Pallas, and (31) Euphrosyne in the S1, S4, first mean value, and first peak integrations.

Asteroid Simulation Number of encounters N(|da|>3σnoise) σa[1]×105 γ1 γ2 T

(10) Hygiea S1 4484 558 71.2 2.4 36.1 0.48

(10) Hygiea S4 4348 551 109.8 10.2 246.1 0.94

(10) Hygiea S6 4410 543 107.3 –9.3 463.1 0.52

(10) Hygiea S9 4422 543 74.8 –1.6 108.8 0.71

(10) Hygiea All (4416±56) (548±7) (90.8±20.6) (0.4±8.1) (213.5±187.8) (0.66±0.21)

(2) Pallas S1 65 1 14.6 –2.3 11.0 0.50

(2) Pallas S4 57 1 21.9 –2.4 10.3 0.61

(2) Pallas S6 69 2 24.2 –4.8 30.5 0.56

(2) Pallas S9 60 3 22.5 –3.4 12.3 0.63

(2) Pallas All (62±5) (2±1) (20.8±4.2) (−3.2±1.1) (16.0±9.7) (0.57±0.05)

(31) Euphrosyne S1 459 92 61.7 3.7 36.3 0.50

(31) Euphrosyne S5 606 138 92.5 2.9 20.4 0.54

(31) Euphrosyne S6 474 113 71.8 –1.2 12.7 0.86

(31) Euphrosyne S7 434 82 65.5 –10.0 12.8 0.60

(31) Euphrosyne All (493±77) (106±24) (72.9±13.7) (−1.2±6.3) (20.6±11.1) (0.63±0.16)

Quite surprisingly, considering the results for (2) Pallas, we found a rather high value of mean σa[1] (72.9 ×10−5 _AU),

which is higher than the mean value for (4) Vesta in Carruba et al. (2012), but lower than the mean value for (1) Ceres in the same article. This unexpected result shows that there is no sim-ple correlation between high inclination and drift rates by close encounters in properaand that values of drift rates need to be investigated with a case-by-case approach.

5. Completeness of the frequency distribution functions

In the previous section, we obtained several distributions of changes in propera(frequency distribution functions hereafter) for diﬀerent massive asteroids and integration schemes. As dis-cussed in Sect. 4, we do observe fluctuations in the numbers of encounters and in the moments of the distributions. A ques-tion that may now arise is how far from obtaining a statistically complete sample our results are. Are our frequency distribution functions (orfdf) of∆avalues a good approximation of the real probability distribution function (pd f hereafter), or is a larger sample of close encounters needed to obtain a more complete statistics? And if a larger sample is needed, what could be the minimum number of close encounters needed to obtain a good approximation of thepd fat, for example, a 3σlevel3_.

To test how complete our frequency distribution functions are in the interval of ∆a changes that we are more interested in (0.0006 < _|∆a_| < 0.006 AU, i.e., between 3σnoise values in

properaand ten times this value), we performed Kolmogorod-Smirnoﬀprobability tests (KS tests hereafter) for each of the ob-served distributions at confidence levels of 1σ, 2σ, and 3σ, with probabilities for the distributions to be compatible of 68.27%, 95.34%, and 99.73%, respectively. Figure5 shows histograms of frequency of changes in proper a caused by (10) Hygiea (panel A), (2) Pallas (panel B), and (31) Euphrosyne (panel C) in the various integration schemes that we used in this work. None of the obtained distributions was compatible at a 3σlevel. However, we found that all Hygiea distributions were compati-ble at a 2σlevel in this_|∆a_|interval, all Euphrosyne distributions

3 _By_σ_{we mean here the standard deviation of the normal or Gaussian}

distribution. A 3σlevel means that there is a 99.73% probability that

the two distributions were compatible.

were compatible at 1σlevel (but only the S1 and S5, and the S6 and the S7 were compatible at a 2σlevel), and only the S1 and S6 Pallas distributions were compatible at a 1σlevel. We believe that our results show that we are obtaining a good approximation of the probability distribution function for (10) Hygiea, a decent one for (31) Euphrosyne, and a poor one for (2) Pallas. In our opinion, improving the completeness of ourfdfs, possibly to a 3σlevel, remain a challenge for future works4. But, before do-ing so, we could question what the minimum number of close encounters needed to obtain a more complete statistics could be. To try answering this question, we turn our attention to the work of Greenberg (1982). In that work, the author computed the change of heliocentric velocity,∆U0, caused by a close

en-counter with a massive body as a function of distance at closest approachrmin and relative velocity with respect to the massive

asteroid V. In Greenberg’s model5 _{the sine of the angle} _χ _at which the asteroid orbit will be deflected due to the close en-counter is given by

sinχ=

⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣1+

rmin·V2

G(M+m)

2⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦

−12

, (4)

whereGis the gravitational constant,Mthe mass of the massive body, andmthe (assumed negligible) mass of the incoming as-teroid. The change in heliocentric velocity∆U0is then given by

∆U0 =

_M

M+m

V

sin 2χd+(cos 2χ₋1)V_,

(5) whered_{is the unit vector pointing in the direction of}rminandV

is the unit vector pointing in the direction ofV. Figure6shows a contour plot of changes in heliocentric velocities as a function ofrminandVfor (10) Hygiea (for brevity we do not show

anal-ogous plots for (2) Pallas and (31) Euphrosyne). The encounters

4 _{In the case of (2) Pallas, one may question the merit of improving our}

knowledge of the details of an eﬀect that is quite certainly fairly small.

5 _{This model is based on a “patching conics” approach, where the}

(7)

−6 −4 −2 0 2 4 6 x 10−3

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Change in a [Au]

Frequency of changes in a

(10) Hygiea

S1 S4 S6 S9

−1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1

x 10−3 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2

Change in a [Au]

(2) Pallas

S1 S4 S6 S9

−6 −4 −2 0 2 4 6

x 10−3

0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05

Change in a [Au]

(31) Euphrosyne

S1 S5 S6 S9

Fig. 5. Histograms of frequency of changes in proper a caused by (10) Hygiea (panel A), (2) Pallas (panel B) and (31) Euphrosyne (panel C) in the integration schemes listed in Table1.

at low relative velocities and low minimum distances cause the maximum change in heliocentric velocities, and, as discussed in Carruba et al. (2007), the maximum change in heliocentric ve-locity is of the order of the escape veve-locity from the massive body (of the order of 200 m/s for bodies of the masses and sizes of (2) Pallas, (4) Vesta, and (10) Hygiea).

Since here we are interested in determining the mini-mum number of close encounters needed to obtain a fairly completefdf in propera, we used the Gauss equation

∆a a =

2

na(1−e2₎1/2

(1+ecosf)∆VT +(esinf)∆VR, (6)

20 40 60 80 100 120 140

2 4 6 8 10 12 14 16 18

Minimum relative distance r

min⋅ 10 3

[km]

Relative velocity V [km/s]

−2 −2 −1.5 −1.5 −1.5 −1.5 −1 −1 −1 −1 −1 −0.5 −0.5 −0.5 −0.5 _−0.5 0 0 0 0 ₀ 0 5 0 5 1 −2 −1.5 −1 −0.5 0 0.5 1 1.5 2

Fig. 6. Contour plot of the log₁₀ of changes in heliocentric veloci-ties∆U0as a function ofrminandVfor (10) Hygiea according to Eq. (5).

where∆VT,∆VR, (and∆VW) are the tangential, radial, and per-pendicular to the orbital plane components of the change in he-liocentric velocity∆U0, n is the minor body mean-motion,a

and e its semi-major axis and eccentricity, and f is the true anomaly at the instant of close encounter. With the simplistic assumptions that the eccentricity of the minor body is zero and that all the change in velocity goes into the tangential compo-nent (this last hypothesis may introduce an error of a factor√3), Eq. (6) reduces to

∆a=2

n∆U0. (7)

We can then use Greenberg’s model to compute maximum changes in properafor diﬀerent values ofrmin andV. Figure7

shows frequency of∆a values computed with Eqs. (7) and (5) for diﬀerent, equally spaced, numbers of close encounters with (10) Hygiea at diﬀerent and equally spaced values ofrminandV

(results are similar for (2) Pallas and (31) Euphrosyne). The val-ues ofrminandVused in Eq. (5) are in the same range of those

shown in Fig.6. Standard KS tests show that the distributions start to be compatible for_≃3000 equally spaced encounters (we tested distributions for up to 6 million encounters with similar re-sults), which sets a lower limit on the minimum number of close encounters that needs to be tested to obtain a fairly completefdf

and that seems to be in agreement with what we found numeri-cally for the simulations with (10) Hygiea. As a further numer-ical test, we also computed the standard deviation of changes inaobtained with Greenberg’s model for 600, 1500, 3000, and 60 000 close encounters. Since this approach only gives abso-lute values of changes ina, the standard deviation of∆adoes not correspond to drift rates, so we computed the variance of the distributions, rescaled to a mean equal to zero, using this formula

σ2_a= 1

N

i=1

(∆a₋0)2= 1

N

i=1

|∆a_|2, (8)

whereN is the number of close encounters. Here we assumed that the mean value of the distributions of changes inawas 0, which seems to be confirmed by our results of Sect.4, where the mean values were all less than 10−6_{AU, i.e., less than the}

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1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 x 10−3 0

0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009

0.01 #of encounters

Change in proper a [AU]

600 1500 3000 60000

Fig. 7.Histogram of ∆achanges computed using Greenberg (1982)

model for diﬀerent numbers of close encounters with (10) Hygiea.

of the rescaled variance given by Eq. (8). To more easily com-pare these results with those of Sect.4, we also multiplied them by 105_{. We obtained values of drift rates of 230.7, 180.1, 127.4,}

and 102.6_×105_{AU for the four distributions, with the last two}

values in the range of what was found in Sect.4for the actual simulations with symplectic integrators. KS tests for values of

∆a>3σnoiseshow that the distributions for 3000 and 60 000

en-ncounters are compatible between them, but not with those with 600 and 1500 encounters. Increasing the number of observed close encounters may, of course, improve the completeness of thefdf. Obtaining better fits for thepd f of (10) Hygiea and (31) Euphrosyne remains a challenge for future works.

6. Close encounters with massive asteroids when the Yarkovsky and YORP effects are considered

In the previous sections, we analyzed the orbital diffusion caused by close encounters with massive asteroids when non-gravitational forces such as the Yarkovsky and YORP effects are not considered. Here we further investigate the subject when these effects are taken into account. For this purpose, we use the São Paulo (SP) integrator, which is able to simulate both close encounters with massive asteroids and the Yarkovsky ef-fect. See Carruba et al. (2007) for a treatment of the integrator and of the Yarkovsky effect. Here we just point out that we used the Vokrouhlický model (Vokrouhlický 1999) for the diurnal and seasonal versions of the Yarkovsky effect, so that the drift rate in semi-major axis is given by

da

dt =k1cosǫ+k2sin

2_ǫ,

(9) wherek1 and k2 are functions depending on the surface

ther-mal parameters and size, andǫ is the spin axis obliquity (see Vokrouhlický 1999, for the exact expressions of the k1 and

k2 functions). In this work, we integrated the same real

aster-oids with H < 13.5 in the regions of the Hygiea, Pallas, and Euphrosyne families considered in the previous sections over 30 Myr with two spin obliquities, ǫ = 0◦ _and _ǫ ₌ ₁₈₀◦_, which maximizes the drift caused by the diurnal version of the Yarkovsky eﬀect. Using Eq. (1), we computed the radius of each object using its absolute magnitude and a geometric albedo equal

Table 3.Number of encounters,σa[1]×105, and Hurts exponentsTfor clones of realH<13.5 asteroids.

Asteroid Spin Nenc σa[1]×105 T

id. orientation [AU]

(10) ǫ=0◦ ₄₈₈₀ ₍₇₀_.₈_±₁₆_.₃₎ ₍₀_.₈₃+1.16

−0.83)

(10) ǫ=180◦ ₄₇₁₉ ₍₈₁_.₈_±₁₈_.₈₎ ₍₀_.₈₁+0.84

−0.81)

(10) All 9599 (76.3±17.5) (0.82±0.64)

(2) ǫ=0◦ ₇₁ ₍₇_.₉_±₁_.₆₎ ₍₀_.₆₃+1.02

−0.63)

(2) ǫ=180◦ ₆₆ ₍₇_.₅_±₁_.₅₎ ₍₀_.₅₆+0.74

−0.56)

(2) All 137 (7.7±1.5) (0.60±0.57)

(31) ǫ=0◦ ₅₀₈ ₍₃₉_.₁_±₇_.₃₎ ₍₀_.₆₅+0.95

−0.65)

(31) ǫ=180◦ ₅₀₇ ₍₅₁_.₅_±₉_.₇₎ ₍₀_.₅₅+0.69

−0.55)

(31) All 1015 (45.3±8.5) (0.60±0.53)

to 0.1, a value typical for B- and C-type objects, the most com-monly found asteroids in the regions of (2) Pallas, (10) Hygiea, and (31) Euphrosyne. We used Yarkovsky parameters that are also typical for B- and C-type objects: a thermal conductivity

K = 0.001 W/m/K (Delbo et al. 2007), a thermal capacity of 680 J/kg/K, a surface density of 1500 kg/m3_{, a bulk density of}

1500 kg/m3_{, a Bond albedo of 0.1, and a thermal emissivity}

of 0.95 (see also Carruba et al.2003, for a more in-depth dis-cussion of these parameters).

Following the approach of Delisle & Laskar (2012), we as-sumed that the obliquity remains constant during a YORP cycle, with maximal values ofǫ=0◦_or_ǫ₌₁₈₀◦_{, which yields a} max-imum strength for the diurnal version of the Yarkovsky eﬀect and a minimal strength for the seasonal one. We also assumed that re-orientations act almost instantaneously at the end of each YORP cycle, with an assumed timescale of 30 Myr for km-sized objects (Delisle & Laskar 2012). We integrated our real asteroids using our S1 and S0 integration schemes (the latter was used to estimate the values ofσnoise, as discussed in the previous

sec-tion), and we used the values of the standard deviations on the moments obtained in the previous section to estimate the possi-ble errors associated with the particular integration scheme used in this work.

As in Sect. 4, we used the three values ofσa[1](t) computed at each 10 Myr to determine theT value of the Hurst exponent. However, with respect to previous works, we also account for the uncertainty associated with changes inσa[1](t) between integra-tions with diﬀerent schemes in the following manner: assuming that errors inσa[1](t) are of the order of the standard deviations found in Sect.4 for (10) Hygiea, (2) Pallas, (31) Euphrosyne, i.e. 22.7%, 20.0%, and 18.8% of the mean value, respectively, we can estimate the value ofT and its error using standard tech-niques of linear regression (see Press et al.2001, Eqs. (15.2.4), (15.2.6), and (15.2.9)).

Table 3 shows values of number of encounters Nenc, of σa[1] × 105 _{AU, and Hurts exponents} _T _{for (10) Hygiea,}

(2) Pallas, and (31) Euphrosyne for (i) the simulation with clones of real asteroids with initial zero obliquity; (ii) the simulation with clones of real asteroids withǫ=180◦_{; and iii) for all} aster-oid clones. For the sake of brevity, we do not report the other moments γ1 and γ2 listed in Table 2. Errors on σa[1]×105

are assumed of the order of what was found in Sect.4. Values ofσa[1]×105 _{for the simulations with the São Paulo}

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considerably lower than the completeness level of≃3000 en-counters found in Sect.5for thefdfto converge. Also, while the

Tvalues that best fitted our data are still compatible with scatter-ing with massive asteroids bescatter-ing a persistent and correlated pro-cess, we unfortunately confirm that, as found in Carruba et al. (2012), the uncertainties are so large that no final conclusions can be positively achieved.

7. Conclusions

We studied the problem of orbital diﬀusion in semi-major axis of minor bodies for the case of encounters with (10) Hygiea, (2) Pallas, and (31) Euphrosyne. Our main results can be summarized as follows:

– We obtained the proper frequencies of precession of the ar-gument of pericenter g and of longitude of the node sfor the three massive asteroids when other massive asteroids (up to 38) are considered along with the eight planets. The values fluctuated beyond a 3σconfidence level for various integra-tion schemes, such as the S9 and S6. Lower fluctuaintegra-tions were also observed when diﬀerent numbers of massive asteroids were considered.

– We investigated the dynamical mobility caused by close en-counters with massive asteroids for real asteroids in the or-bital regions of (10) Hygiea, (2) Pallas, and (31) Euphrosyne with absolute magnitude H < 13.5. As in Carruba et al. (2012), we found that not only diﬀerent asteroids experi-enced close encounters with the three massive bodies when diﬀerent integration schemes were considered, but also that the whole statistics of close encounters fluctuated. Variances of the change in properamassive asteroids varied up to 23% in the four integration schemes that we used. We confirm the hypothesis of Bottke et al. (1994) that encounters with highly inclined objects as (2) Pallas are indeed a minor ef-fect because of rather high relative speed and distances at encounters between perturber and perturbee. Unexpectedly, we found a very large mean value of drift rate in propera

caused by close encounters with (10) Hygiea,σa[1]×105=

(90.8 ±20.6) AU, and (31) Euphrosyne, σa[1]×105 ₌

(72.9±13.7) AU. These values are higher than the drift rate caused by (4) Vesta for its orbital region (σa[1]×105 ₌

(29.5±10.7) AU), as found in Carruba et al. (2012), and may have repercussion on the orbital evolution of members of their respective families.

– We performed KS tests on the completeness of thefdf ob-tained for the three massive asteroids. The frequency distri-butions obtained with the diﬀerent integration schemes are compatible among themselves to a 2σlevel for (10) Hygiea, to a 1σlevel for (31) Euphrosyne, and to lower levels for (2) Pallas. Considerations based on the model of Greenberg (1982) suggest that the minimum number of close encoun-ters needed to obtain a good fit of the pd f are of the or-der of 3000, with expected uncertainties on the measured drift rates of the order of 10% for this number of encoun-ters. Obtaining better fits of thepd ffor (10) Hygiea and (31) Euphrosyne, possibly to a 3σlevel, remains a challenge for future works.

– We studied the eﬀect of close encounters with massive asteroids when the Yarkovsky and YORP eﬀects were also considered and computed the value of the Hurst exponentT

of the diﬀusion as a function of time, including the eﬀect

of the uncertainty caused by differences in the integration schemes as found in this work. Drift rate values and Hurst exponents for the Hygiea region are in the same range, to within the errors, as those found in the conservative integra-tions (for the Pallas and Euphrosyne areas, they are some-what smaller, but this may be due to the limited number of close encounters that occurred during the simulations for as-teroids in these regions), and Hurst exponents indicate scat-tering with massive asteroids being a persistent and corre-lated process (0.5 < T < 1.0, but values are affected by large uncertainties). This suggests that in order to obtain es-timates of these parameters on the timescales used in this work (30 Myr), including the Yarkovsky and YORP effects may not be necessary.

In this work we investigated for the first time the long-term eﬀect that close encounters with massive asteroids such (10) Hygiea, (2) Pallas, and (31) Euphrosyne may have had on the orbital evo-lution of asteroids in their orbital proximity. In particular, the un-expected high rate of drift in properacaused by encounters with (10) Hygiea and (31) Euphrosyne, unknown in previous works in the literature, suggests that this diﬀusion mechanism may have played a significant role in the evolution of their respective fami-lies that is yet to be investigated. Also, the relative large value of drift rate found for (31) Euphrosyne shows that there is no sim-ple correlation between the high inclination of the perturber and the drift rates caused by close encounters. According to our re-sults, values of drift rates need therefore to be investigated with a case-by-case approach.

Acknowledgements. We are grateful to the anonymous referee of this paper for comments and suggestions that significantly improved the quality of our work. We also thank G. Valsecchi for discussions and hints that opened interesting new lines of research. This work was supported by the Brazilian National Research Council (CNPq), project 305453/2011-4, by the Foundation for Supporting Scientific Research in the São Paulo state (FAPESP), grant 11/19863-3, and by the Coordenation for Improvement of Higher Education Personnel (CAPES), grant 23038.007093/2012-13.

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Appendix A: Asteroid astrometric masses

We report in TableA.1the identification of each asteroid used in this work, its mass and mass uncertainty as in Carry (2012), and the percentile

errors associated with each asteroid mass (see Carry2012, for a discussion of the methods used to determine the asteroid masses, and their

uncertainties).

Table A.1.Asteroid astrometric masses with their uncertainties.

Asteroid Mass (in 1015_kg) _{Mass uncertainty (in 10}15_kg) _{Percentile error}

1 Ceres 944 000. 6000. 0.006

4 Vesta 259 000. 0. 0.000

2 Pallas 204 000. 4000. 0.020

10 Hygiea 86 300. 5200. 0.060

511 Davida 33 800. 10 200. 0.302

704 Interamnia 32 800. 4500. 0.137

15 Eunomia 31 400. 1800. 0.057

3 Juno 27 300. 2900. 0.106

16 Psyche 27 200. 7500. 0.276

536 Merapi 26 100. 4700. 0.180

52 Europa 23 800. 5800. 0.244

165 Loreley 19 100. 1900. 0.099

88 Thisbe 15 300. 3100. 0.203

87 Sylvia 14 800. 0. 0.000

420 Bertholda 14 800. 900. 0.061

6 Hebe 13 900. 1000. 0.072

65 Cybele 13 600. 3100. 0.228

212 Medea 13 200. 1000. 0.076

7 Iris 12 900. 2100. 0.163

29 Amphitrite 12 900. 2000. 0.155

690 Wratislavia 12 800. 300. 0.023

31 Euphrosyne 12 700. 6500. 0.512

57 Mnemosyne 12 600. 2400. 0.190

147 Protogeneia 12 300. 500. 0.041

675 Ludmilla 12 000. 2400. 0.200

409 Aspasia 11 800. 2300. 0.195

532 Herculina 11 500. 2800. 0.243

107 Camilla 11 200. 300. 0.027

451 Patientia 10 900. 5300. 0.486

200 Dynamene 10 700. 1600. 0.150

444 Gyptis 10 600. 2800. 0.264

324 Bamberga 10 300. 1000. 0.097

602 Marianna 10 200. 500. 0.049

895 Helio 9870. 6050. 0.613

154 Bertha 9190. 5200. 0.566

381 Myrrha 9180. 800. 0.087

8 Flora 9170. 1750. 0.191

13 Egeria 8820. 4250. 0.482